This paper is devoted to the computation of Lyapunov quantities (Poincare-Lyapunov constants) and to the study of limit cycles of two-dimensional dynamical systems. Here new method for computation of Lyapunov quantities is suggested, which is based on the constructing approximations of solution in the original Euclidean coordinates and in the time domain. The advantages of this method are in its ideological simplicity and visualization power. This approach can also be applied to the problem of distinguishing of isochronous center. New method of asymptotic integration for Lienard equation is suggested which is effective for localization of attractors and investigation of large limit cycles. This technique is applied for investigation of autonomous quadratic systems. The quadratic system is reduced to the Lienard equation and then the two-dimensional domain of parameters, corresponding to the existence of four limit cycles (three small and one large), is evaluated. This domain extends the domain of parameters, obtained for quadratic system with four limit cycles by S.L. Shi in 1980.